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Generation of a Gaussian Schell-model field as a mixture of its coherentmodesTo cite this article: Abhinandan Bhattacharjee et al 2019 J. Opt. 21 105601

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Generation of a Gaussian Schell-model fieldas a mixture of its coherent modes

Abhinandan Bhattacharjee , Rishabh Sahu and Anand K Jha

Department of Physics, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India

E-mail: akjha9@gmail.com

Received 26 April 2019, revised 15 July 2019Accepted for publication 14 August 2019Published 4 September 2019

AbstractGaussian Schell-model (GSM) fields are examples of spatially partially coherent fields which inrecent years have found several unique applications. The existing techniques for generatingGSM fields are based on introducing randomness in a spatially completely coherent field and arelimited in terms of control and precision with which these fields can be generated. In contrast, wedemonstrate an experimental technique that is based on the coherent mode representation ofGSM fields. By generating individual coherent eigenmodes using a spatial light modulator andincoherently mixing them in a proportion fixed by their normalized eigenspectrum, weexperimentally produce several different GSM fields. Since our technique involves only theincoherent mixing of coherent eigenmodes and does not involve introducing any additionalrandomness, it provides better control and precision with which GSM fields with a given set ofparameters can be generated.

Keywords: optical coherence, interferometry, measurement, classical optics

(Some figures may appear in colour only in the online journal)

1. Introduction

Spatially partially coherent fields have been extensively stu-died in the past few decades, and among such fields, theGaussian Schell-model (GSM) field has been the mostimportant [1–6]. This is because GSM fields are widely usedin theoretical models due to their simple functional form andhave found several unique applications in areas including freespace optical communication [7, 8], ghost imaging [9, 10],propagation through random media and atmospheric condi-tions [11–13], particle trapping [14], and optical scattering[15]. A GSM field is characterized by a Gaussian transverseintensity profile and a Gaussian degree of coherence function.The widths of these Gaussian functions are the parametersthat characterize a GSM field. There are several experimentaltechniques for generating GSM fields [16–20]. In all thesetechniques, partial spatial coherence is generated by intro-ducing randomness in a spatially completely coherent Gaus-sian beam and then by ensuring that the transverse intensityprofile of the randomized field stays Gaussian. The mostcommon way of introducing randomness is by using arotating ground glass plate (RGGP), which causes the perfect

spatial correlation of the incoming field to reduce to aGaussian correlation. The other ways of introducing ran-domness include using either an acousto-optic modulator or aspatial light modulator (SLM) [21–23].

Although the above mentioned experimental techniquesensure that the transverse intensity, as well as the degree ofcoherence of the generated field, are Gaussian functions, thesetechniques are limited in terms of precision and control.Therefore, an efficient experimental technique with precisecontrol for generating a GSM field is still required. In thisarticle, we demonstrate just such a technique, which, incontrast to the techniques mentioned above, does not expli-citly involve introducing additional randomness. Our techni-que is based on the coherent mode representation of GSMfields [24–26]. The coherent mode representation is the wayof describing a spatially partially coherent field as a mixtureof several spatially completely coherent modes. Therefore, aGSM field with any given set of parameters can be generatedwith precision by first producing different coherent modesand then incoherently mixing them in a proportion fixed bythe coherent mode representation of the field.

Journal of Optics

J. Opt. 21 (2019) 105601 (7pp) https://doi.org/10.1088/2040-8986/ab3b24

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https://orcid.org/0000-0002-0646-2404https://orcid.org/0000-0002-0646-2404mailto:akjha9@gmail.comhttps://doi.org/10.1088/2040-8986/ab3b24https://crossmark.crossref.org/dialog/?doi=10.1088/2040-8986/ab3b24&domain=pdf&date_stamp=2019-09-04https://crossmark.crossref.org/dialog/?doi=10.1088/2040-8986/ab3b24&domain=pdf&date_stamp=2019-09-04

2. Theory

2.1. GSM field as a mixture of its constituent coherent modes

The coherent mode representation of a one-dimensional GSMfield was worked out in [24–26]. Here, we present it for thetwo-dimensional case. The cross spectral density of a GSMfield is given by

r r r r r rm= -W I I, 11 2 1 2 1 2( ) ( ) ( ) ( ) ( )

with r r s= -I A exp 2 s1 2 12 2( ) [ ( )], r r s= -I A exp 2 s2 2 2

2 2( ) [ ( )]and r rm r s- = - Dexp 2 g1 2 2

2( ) [ ( ) ( )]. Here, r º x y,1 1 1( ),r º x y,2 2 2( ), r rr r= =,1 21 2∣ ∣ ∣ ∣, r rrD = -1 2∣ ∣, and A isa constant. rI 1( ) is the intensity at point r1 with σs being therms width of the beam, and r rm -1 2( ) is the degree ofspatial coherence between points r1 and r2, with σg being therms spatial coherence width of the beam. r rW ,1 2( ) can bewritten in terms of its coherent mode representation as [24]:

r r r rl= å åW W, ,m n mn mn1 2 1 2( ) ( ), where λmn are theeigenvalues and r r r rf f=W ,mn mn mn1 2 1 2*( ) ( ) ( ) are the cross-spectral density functions corresponding to the coherenteigenmodes rfmn ( ). Following [24], we write equation (1) as

åår r l f f=W x y x y, , , , 2m n

mn mn mn1 2 1 1 2 2*( ) ( ) ( ) ( )

where

lp

fp

=+ + + +

=

´

+

+

- +

Aa b c

b

a b c

x yc

m n

H x c H y c e

,2 1

2

2 2 .

mn

m n

mn m n

m nc x y

2

12

2 2

⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠( ) ! !

( ) ( )

( )

( )

We have s s= = = +a b c a ab1 4 , 1 2 , 2s g2 2 2 1 2( ) ( ) ( ) ,

and Hm(x) are the Hermite polynomials. Substitutingσg/ σs=q, we write λmn as

l l=+ + +

+

q q q

1

2 1 2 1. 3mn

m n

00 2 2 1 2

⎡⎣⎢

⎤⎦⎥( ) [( ) ] ( )

( )

The quantity q is a measure of the ‘global degree ofcoherence’ of the field. For fixed σs, higher values of q implyhigher values for the degree of spatial coherence. In whatfollows, it will be very convenient to work with the normal-ized eigenvalues lmn¯ . So, for that purpose, we first takeλ00=1 and then define lmn¯ as: l l l= åmn mn mn mn¯ ( ) suchthat lå = 1mn mn¯ .

The coherent mode representation describes a partiallycoherent field as an incoherent mixture of completelyuncorrelated coherent modes. We note that equations (1) and(2) are the two equivalent descriptions of the same cross-spectral density function r rW ,1 2( ) for the GSM field. Whileequation (1) describes r rW ,1 2( ) as a single partially coherentfield, equation (2) describes it as an incoherent mixture ofspatially completely coherent eigenmodes fmn(x, y) with theirproportions given by the normalized eigenvalues lmn¯ . Theintrinsic randomness of the GSM field, which is described byequation (1) as being across the transverse plane of the field,gets described by equation (2) as complete randomnessbetween different coherent eigenmodes. Equation (2) showsthat to generate a GSM field, one needs to generate the spa-tially completely coherent eigenmodes fmn(x, y) and then mixthem incoherently in lmn¯ proportion. We also find that for anormalized eigenspectrum, the coherent mode representationof equation (2) has only q and c as free parameters. Parameterq decides the exact proportion lmn¯ of the eigenmodesfmn(x, y) and the parameter c decides the overall transverseextent of the field. Thus, by controlling q and c, one cangenerate any desired GSM field.

The coherent mode representation of a GSM field havingonly one term represents a completely coherent Gaussian fieldwith s = ¥g . On the other hand, a completely incoherentGSM field implies σg→0, and in this limit, the coherentmode representation contains an infinite number of terms.When σg is finite, the field is partially coherent, and thecoherent mode representation contains only a finite number ofterms with significant eigenvalues. Figure 1 shows thetheoretical plots of normalized eigenvalues lmn¯ for threedifferent values of q, namely, q=0.8, q=0.5, andq=0.25. The value of c for all the fields is 1.34 mm−2. Wefind that to generate GSM fields with the smaller globaldegree of coherence q one requires to mix a larger number ofeigenmodes.

Figure 1. Theoretical plots of the normalized eigenvalues lmn¯ for three different values of the degree of global coherence, namely, forq=0.80, q=0.50, and q=0.25.

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2.2. Measuring GSM fields

For measuring the cross-spectral density of a GSM field, weuse the measurement technique of [27]. This is a two-shottechnique involving wavefront-inversion inside an inter-ferometer and is the spatial analog of the technique [28]recently demonstrated for measuring the angular coherencefunction [29]. Figure 2(b) shows the schematic diagram ofthe measurement technique. For a GSM field input, theintensity rIout ( ) at the output port of the interferometer is givenby r r r r r d= + - + -I k I k I k k W2 , cosout 1 2 1 2( ) ( ) ( ) ( )

[27]. Here k1 and k2 are the scaling constants in the two arms andδ is the overall phase difference between the two interferometricarms; rI ( ) is the intensity of the GSM field at point r and

r r-W ,( ) is the cross spectral density of the GSM field for thepair of points r and r- . Now, suppose there are two outputinterferograms with intensities rdIoutc¯ ( ) and rdIoutd¯ ( ) measured atδ=δc and δ=δd, respectively. As worked out in [27], if theshot-to-shot variation in the background intensity is negligible,the difference r r rD = -d dI I Iout out outc d¯ ( ) ¯ ( ) ¯ ( ) in the intensitiesof the two interferograms is given by rD =I k k2out 1 2¯ ( )

Figure 2. (a) Schematic setup for generating GSM fields. (b) Schematic setup for measuring the cross-spectral density function using the techniqueof [27]. Here, we have SLM: spatial light modulator; BS: beam splitter; M: mirror; and L: converging lens. (c) The theoretically expected andexperimentally generated intensity corresponding to the eigenmodes f11(x, y), f44(x, y), and f77(x, y).

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J. Opt. 21 (2019) 105601 A Bhattacharjee et al

r rd d- -Wcos cos ,c d( ) ( ). We find that the difference inten-sity is proportional to the cross-spectral density function rW ,(r- ). Using equation (1), we write r r r- =W W, 2( ) ( )

W x y2 , 2( ) = - -rs

rs

A exp exp2 28

2

2s g

2

2

2

2

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

( ) ( ) , and therefore we

get

r rD µI W 2 4out¯ ( ) ( ) ( )

that is, the difference intensity rDIout¯ ( ) is proportional torW 2( ). Thus by measuring the difference intensity the cross-

spectral density function rW 2( ) can be directly measuredwithout having to know k1, k2 and δ precisely. Using rW 2( )andrI ( ), the degree of coherence m r2( ) of the field can be written as

r rr r

rr

m =-

=W

I I

W

I2

2 2, 5( ) ( )

( ) ( )( )( )

( )

where r r= -I I( ) ( ) because the transverse intensity profile ofa GSM field is symmetric about inversion.

3. Experimental results

3.1. Generation of GSM field

Figures 2(a) and (b) show the experimental setup for generatingthe GSM field and measuring its cross-spectral density functionin a two-shot manner [27], respectively. The Gaussian field froma 5mW He-Ne laser is incident on a Holoeye Pluto SLM and anappropriate phase pattern is displayed on the SLM in order togenerate a given eigenmode at the detection plane of theEMCCD camera. In particular, the SLM is programmed togenerate different eigenmodes fmn(x, y) using the Arrizón

method [30]. Figure 2(c) shows the experimentally measured andtheoretically expected intensity profiles of eigenmodes: f11(x, y),f44(x, y), and f77(x, y). We find a good match between the theoryand experiment. Now, in order to produce a GSM field with agiven q, that is, a given eigenspectrum lmn¯ , we need to producethe incoherent mixture of different eigenmodes with proportiongiven bylmn¯ . It is done in the following manner. First, the phasepatterns corresponding to different eigenmodes are displayed onthe SLM sequentially. The weights lmn¯ are fixed by making thedisplay-time of the phase pattern corresponding to an eigenmodefmn(x, y) proportional to the corresponding eigenvalue lmn¯ . Inour experiment, the SLM works at 60 Hz. The display-time of agiven phase pattern on the SLM is of the order of tens of mil-liseconds while the coherence time of our He–Ne laser is in tensof picoseconds. Although the SLM introduces a deterministicphase modulation along the beam cross-section for a giveneigenmode, the phase modulation for a given eigenmode iscompletely uncorrelated with that for any other eigenmode. Inthis way, as long as the time of observation is kept long enoughfor all the modes to get detected, the SLM produces an inco-herent mixture of coherent modes. Therefore, the exposure timeof the EMCCD camera is made equal to the total display-time ofall the phase patterns such that the camera collects all the gen-erated eigenmodes.

Using the procedure described above, we generate GSMfields for three different values of q, namely, q=0.8, q=0.5,and q=0.25. Although in principle for any given q we need aninfinite number of modes to produce the corresponding GSMfield precisely. However, the plots in figure 1 show that for anyfinite q the number of eigenvalues lmn¯ with significant con-tributions are only finite and that the number of significanteigenvalues increases with decreasing q. In our experiment, wekeep l´0.07 00¯ as the cutoff for deciding the eigenmodes witha significant contribution. This means that for a given q wegenerate only those eigenmodes for which l l´ 0.07mn 00¯ ¯ .With this cutoff, we generate 10, 21, and 66 eigenmodes,respectively, for the three values of q. The sum of these eigen-values låmn mn¯ turns out to be about 0.87, 0.84, and 0.82,respectively, for the three q values, which are quite close to one.

3.2. Measurement of the field

Each of the generated GSM fields is made incident on theinterferometer in figure 2(b). In our experiment, the SLM worksat 60 Hz, and the EMCCD camera was kept open for 1.40, 3.00,and 5.76 seconds, respectively, for the three q values. This wasfor ensuring that the camera collects all the generated eigen-modes. The value of c in each case was 1.34mm−2. For eachvalue of q, we collect two interferograms, one with δ=δc≈0and the other one with δ=δd≈π. These two interferograms arethen subtracted to generate the difference intensity rDIout¯ ( ),which is then scaled such that the value of the most intense pixelis equal to one. From equation (4), we have that the scaleddifference intensity rDIout¯ ( ) is nothing but the scaled cross-spectral density function r =W W x y2 2 , 2( ) ( ). To ensure thatthe interferograms are not drifting and that the shot-to-shotbackground intensity variation is negligible, we cover the entireinterferometer with a box. Figures 3(a), (d), and (g) show the

Figure 3. Plots of the the cross-spectral density function of GSMfields with q=0.8, q=0.5,and q=0.25. For the three values of q,(a), (d) and (g) are the experimentally measured cross-spectraldensity functions W(2x, 2y) while (b), (e) and (h) are thecorresponding theoretical plots. (c), (f) and (i) The plots of the one-dimensional cuts along the x-direction of the theoretical andexperimental cross-spectral density functions.

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J. Opt. 21 (2019) 105601 A Bhattacharjee et al

experimentally measured cross-spectral density functions for thethree values of q while figures 3(b), (e), and (h) show thecorresponding theoretical cross-spectral density functions plottedusing equation (1). In order to compare our experimental resultswith the theory, we take the one-dimensional cuts of the theor-etical and experimental cross-spectral density functions and plotthem together in figures 3(c), (f), and (i), for the three values of q.

Next, we measure the transverse intensity profile of theGSM field for different values of q. For measuring theintensity profile, we block the interferometric arm containingthe lens and record the intensity at the EMCCD camera plane.Figures 4(a), (g) and (m) show the measured intensity profilesr =I I x y,( ) ( ) for the three values of q. The corresponding

theoretical intensities as given by equation (1) are plotted infigures 4(b), (h) and (n), respectively. The experimental andtheoretical plots are both scaled such that the value of themost intense pixel is equal to one. Again, for comparing ourexperimental results with theory, we plot in figures 4(c), (i)and (o) the one-dimensional cuts along the x-direction of thetheoretical and experimental intensity profiles.

Finally, using the intensity and cross-spectral densitymeasurements above, we find the degree of coherencem x y2 , 2( ). Figures 4(d), (j) and (p) show the degree ofcoherence functions for the three q values while figures 4(e),(k), and (q) show the corresponding theoretical plots. Wescale both the experimental and theoretical plots such that the

Figure 4. Plots of the intensity and the degree of coherence of GSM fields with q=0.8, q=0.5,and q=0.25. For the three values of q, (a),(g) and (m) show the experimentally measured intensity profiles I(x, y) while (b), (h) and (n) are the corresponding theoretical plots. (c), (i)and (o) The plots of the one-dimensional cuts along the x-direction of the theoretical and experimental intensity profiles. For the three valuesof q, (d), (j) and (p) are the experimentally measured degrees of coherence μ(2x, 2y) while (e), (k) and (q) are the corresponding theoreticalplots. (f), (l) and (r) are the plots of the one-dimensional cuts along the x-direction of the theoretical and experimentally measured degree ofcoherence functions.

Figure 5. Plots of the numerically simulated degree of coherence function for q = 0.25 have been shown for the three values of the cutoff onlmn. For each plot, the solid line represents the theoretical degree of coherence generated using equation (1).

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J. Opt. 21 (2019) 105601 A Bhattacharjee et al

value of the most intense pixel is equal to one. To furthercompare our experimental results with the theory, we take theone-dimensional cuts along the x-direction of the theoreticaland experimental degree of coherence and plot them togetherin figures 4(f), (l) and (r) for the three values of q. The resultsshow that with decreasing q the width of the degree ofcoherence function decreases while the width of the trans-verse intensity profile increases. This is due to the fact that forgenerating fields with large q values, one requires to mixtogether a lower number of eigenmodes, whereas for gen-erating fields with smaller q values, one is required to mixtogether a larger number of eigenmodes, as illustrated infigure 1. We note that although the individual eigenmodes arespatially perfectly coherent, the increase in the number ofeigenmodes in the incoherent mixture increases the random-ness and thereby decreases the width of the degree ofcoherence function, while increasing the transverse width ofthe beam. We find a good match between the theory andexperiment for each value of q.

From the above result, it is evident that the GSM field withq = 0.80 has a better match with the theory than the field withq = 0.25. The reasons for this are as follows. First of all, asillustrated in figure 1, the eigenvalue distribution for q=0.25 isbroader than that for q=0.80. As a result, for producing theGSM field with q=0.25, we need to generate a larger numberof modes with corresponding eigenvalues lmn¯ . As mentionedearlier, the eigenvaluelmn¯ is assigned by the display time of thecorresponding eigenmode fmn(x, y) on the SLM. Now, sincethe refresh rate of our SLM is 60 Hz and the collection time ofthe detection camera is in seconds, we have only a few hundreddiscrete time-bins for assigning the eigenvalueslmn¯ . This puts alimit on the precision with which a given number of modes witheigenvalues lmn¯ could be generated and therefore results in abetter match for GSM field with q = 0.80 since that requires alower number of modes to be produced. Nevertheless, thislimitation can be overcome by using a faster SLM, which canprovide a greater number of time-bins for a given collectiontime and thereby can improve the precision with which lmncould be generated. The other reason for a better match atq = 0.80 is the cutoff on lmn¯ , which restricts the number ofeigenmodes in the incoherent mixture. For the given cutoff of

l0.07 00 on lmn, the sum låmn mn¯ becomes 0.87 and 0.82 forq=0.80 and 0.25, respectively, resulting in a better match forq = 0.80. Figure 5 shows the effect of the cutoff on the pre-cision with which GSM field could be generated. In the figure,we have plotted the numerically simulated degree of coherencefor q= 0.25 for three different values of the cutoffs. We see thatas the cutoff is lowered, a GSM field with a better match withthe theory can be produced. Thus, in our technique, one candecide the cutoff for the eigenvalues depending on the precisionrequirement for a given application.

4. Conclusion and discussion

In conclusion, we have demonstrated in this article an exper-imental technique for generating GSM fields that is based on itscoherent mode representation. We have reported the generation

of GSM fields with a range of values for the global degreeof coherence, and to the best of our knowledge, such ademonstration has not been reported earlier. Compared to theexisting techniques for producing GSM fields [16–23], the mainadvantage of our technique is that it does not explicitly involveintroducing any additional randomness. As a result, the errorsinvolved in our scheme are mostly systematic and are easilycontrollable. This fact has also been highlighted in the recentlydemonstrated techniques [31–33] for producing different typesof spatially partially coherent fields without introducing addi-tional randomness. The other advantage of our method is that itis SLM-based; therefore the phase patterns can be changedelectronically and different GSM fields can be generated by justcontrolling c and q without having to remove any physicalelement from a given experimental setup. This is as opposed towhen producing a partially coherent field using an RGGP, inwhich case one requires separate RGGPs with precisely char-acterized features for producing different GSM fields. Thus, ourmethod provides much better control and precision with whichGSM fields can be generated. We therefore expect our methodto have important practical implications for applications that arebased on utilizing spatially partially coherent fields.

Acknowledgments

We thank Shaurya Aarav for useful discussions and acknowl-edge financial support through the research Grant No. EMR/2015/001931 from the Science and Engineering ResearchBoard (SERB), Department of Science and Technology, Gov-ernment of India.

ORCID iDs

Abhinandan Bhattacharjee https://orcid.org/0000-0002-0646-2404

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1. Introduction2. Theory2.1. GSM field as a mixture of its constituent coherent modes2.2. Measuring GSM fields

3. Experimental results3.1. Generation of GSM field3.2. Measurement of the field

4. Conclusion and discussionAcknowledgmentsReferences